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Chapter 12: Inventory Control

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Title: Chapter 12: Inventory Control


1
Chapter 12 Inventory Control

2
Purposes of Inventory
  • 1. To maintain independence of operations
  • 2. To meet variation in product demand
  • 3. To allow flexibility in production scheduling
  • 4. To provide a safeguard for variation in raw
    material delivery time
  • 5. To take advantage of economic purchase-order
    size

3
Inventory Costs
  • Holding (or carrying) costs
  • Costs for capital, storage, handling,
    shrinkage, insurance, etc
  • Setup (or production change) costs
  • Costs for arranging specific equipment setups,
    etc
  • Ordering costs
  • Costs of someone placing an order, etc
  • Shortage costs
  • Costs of canceling an order, etc

4
Independent vs. Dependent Demand
Finished product
E(1)
Component parts
5
Inventory Systems
  • Single-Period Inventory Model
  • One time purchasing decision (Example vendor
    selling t-shirts at a football game)
  • Seeks to balance the costs of inventory overstock
    and under stock
  • Multi-Period Inventory Models
  • Fixed-Order Quantity Models
  • Event triggered (Example running out of stock)
  • Fixed-Time Period Models
  • Time triggered (Example Monthly sales call by
    sales representative)

6
The Newsvendor Model
7
Wetsuit example
  • The too much/too little problem
  • Order too much and inventory is left over at the
    end of the season
  • Order too little and sales are lost.
  • Example Selling Wetsuits
  • Economics
  • Each suit sells for p 180
  • Seller charges c 110 per suit
  • Discounted suits sell for v 90

8
Too much and too little costs
  • Co overage cost (i.e. order one too many ---
    demand
  • The cost of ordering one more unit than what you
    would have ordered had you known demand if you
    have left over inventory the increase in profit
    you would have enjoyed had you ordered one fewer
    unit.
  • For the example Co Cost Salvage value c
    v 110 90 20
  • Cu underage cost (i.e. order one too few
    demand order amount)
  • The cost of ordering one fewer unit than what you
    would have ordered had you known demand if you
    had lost sales (i.e., you under ordered), Cu is
    the increase in profit you would have enjoyed had
    you ordered one more unit.
  • For the example Cu Price Cost p c 180
    110 70

9
Newsvendor expected profit maximizing order
quantity
  • To maximize expected profit order Q units so that
    the expected loss on the Qth unit equals the
    expected gain on the Qth unit
  • Rearrange terms in the above equation -
  • The ratio Cu / (Co Cu) is called the critical
    ratio (CR).
  • We shall assume demand is distributed as the
    normal distribution with mean m and standard
    deviation s
  • Find the Q that satisfies the above equality use
    NORMSINV(CR) with the critical ratio as the
    probability argument.
  • (Q-m)/s z-score for the CR so
  • Q m z s

Note where F(Q) Probability Demand
10
Finding the examples expected profit maximizing
order quantity
  • Inputs
  • Empirical distribution function table p 180 c
    110 v 90 Cu 180-110 70 Co 110-90
    20
  • Evaluate the critical ratio
  • NORMSINV(.7778) 0.765
  • Other Inputs mean m 3192 standard deviation
    s 1181
  • Convert into an order quantity
  • Q m z s
  • 3192 0.765 1181
  • 4095

Find an order quantity Q such that there is a
77.78 prob that demand is Q or lower.
11
Single Period Model Example
  • A college basketball team is playing in a
    tournament game this weekend. Based on past
    experience they sell on average 2,400 tournament
    shirts with a standard deviation of 350. They
    make 10 on every shirt sold at the game, but
    lose 5 on every shirt not sold. How many shirts
    should be ordered for the game?
  • Cu 10 and Co 5 P 10 / (10 5)
    .667
  • Z.667 .432 (use NORMSINV(.667) therefore we
    need 2,400 .432(350) 2,551 shirts

12
Hotel/Airline Overbooking
  • The forecast for the number of customers that DO
    NOT SHOW UP at a hotel with 118 rooms is Normally
    Dist with mean of 10 and standard deviation of 5
  • Rooms sell for 159 per night
  • The cost of denying a room to the customer with a
    confirmed reservation is 350 in ill-will and
    penalties.
  • Let X be number of people who do not show up X
    follows a probability distribution!
  • How many rooms ( Y ) should be overbooked (sold
    in excess of capacity)?
  • Newsvendor setup
  • Single decision when the number of no-shows in
    uncertain.
  • Underage cost if X Y (insufficient number of
    rooms overbooked).
  • For example, overbook 10 rooms and 15 people do
    not show up lose revenue on 5 rooms
  • Overage cost if X overbooked).
  • Overbook 10 rooms and 5 do not show up pay
    penalty on 5 rooms

13
Overbooking solution
  • Underage cost
  • if X Y then we could have sold X-Y more rooms
  • to be conservative, we could have sold those
    rooms at the low rate, Cu rL 159
  • Overage cost
  • if X
  • and incur an overage cost Co 350 on each
    bumped customer.
  • Optimal overbooking level
  • Critical ratio

14
Optimal overbooking level
  • Suppose distribution of no-shows is normally
    distributed with a mean of 10 and standard
    deviation of 5
  • Critical ratio is
  • z NORMSINV(.3124) -0.4891
  • Y m z s 10 -.4891 5 7.6
  • Overbook by 7.6 or 8
  • Hotel should allow up to 1188 reservations.

15
Multi-Period ModelsFixed-Order Quantity Model
Model Assumptions (Part 1)
  • Demand for the product is constant and uniform
    throughout the period
  • Lead time (time from ordering to receipt) is
    constant
  • Price per unit of product is constant
  • Inventory holding cost is based on average
    inventory
  • Ordering or setup costs are constant
  • All demands for the product will be satisfied (No
    back orders are allowed)

16
Basic Fixed-Order Quantity Model and Reorder
Point Behavior
17
Cost Minimization Goal
By adding the item, holding, and ordering costs
together, we determine the total cost curve,
which in turn is used to find the Qopt inventory
order point that minimizes total costs
C O S T
Holding Costs
Ordering Costs
Order Quantity (Q)
18
Deriving the EOQ
  • Using calculus, we take the first derivative of
    the total cost function with respect to Q, and
    set the derivative (slope) equal to zero, solving
    for the optimized (cost minimized) value of Qopt

We also need a reorder point to tell us when to
place an order
19
Basic Fixed-Order Quantity (EOQ) Model Formula
TCTotal annual cost D Demand C Cost per unit Q
Order quantity S Cost of placing an order or
setup cost R Reorder point L Lead time HAnnual
holding and storage cost per unit of inventory
Total Annual Cost
Annual Purchase Cost
Annual Ordering Cost
Annual Holding Cost


20
EOQ Class Problem 1
  • Dickens Electronics stocks and sells a particular
    brand of PC. It costs the firm 450 each time it
    places and order with the manufacturer. The cost
    of carrying one PC in inventory for a year is
    170. The store manager estimates that total
    annual demand for computers will be 1200 units
    with a constant demand rate throughout the year.
    Orders are received two days after placement
    from a local warehouse maintained by the
    manufacturer. The store policy is to never have
    stockouts. The store is open for business every
    day of the year. Determine the following
  • Optimal order quantity per order.
  • Minimum total annual inventory costs (i.e.
    carrying plus ordering ignore item costs).
  • The optimum number of orders per year (D/Q)

21
EOQ Problem 2
  • The Western Jeans Company purchases denim from
    Cumberland textile Mills. The Western Jeans
    Company uses 35,000 yards of denim per year to
    make jeans. The cost of ordering denim from the
    Textile Mills is 500 per order. It costs
    Western 0.35 per yard annually to hold a yard of
    denim in inventory. Determine the following
  • a. Optimal order quantity per order.
  • b. Minimum total annual inventory costs (i.e.
    carrying plus ordering).
  • c. The optimum number of orders per year.
  •       

22
Problem 3
  • A store specializing in selling wrapping paper is
    analyzing their inventory system. Currently the
    demand for paper is 100 rolls per week, where the
    company operates 50 weeks per year.. Assume that
    demand is constant throughout the year. The
    company estimates it costs 20 to place an order
    and each roll of wrapping paper costs 5.00 and
    the company estimates the yearly cost of holding
    one roll of paper to be 50 of its cost.
  • If the company currently orders 200 rolls every
    other week (i.e., 25 times per year), what are
    its current holding and ordering costs (per
    year)?

23
Problem 3
  • The company is considering implementing an EOQ
    model. If they do this, what would be the new
    order size (round-up to the next highest
    integer)? What is the new cost? How much money
    in ordering and holding costs would be saved each
    relative to their current procedure as specified
    in part a)?
  • The vendor says that if they order only twice per
    year (i.e., order 2500 rolls per order), they can
    save 10 cents on each roll of paper i.e., each
    roll would now cost only 4.90. Should they take
    this deal (i.e., compare with part bs answer)
    Hint For c. calculate the item, holding, and
    ordering costs in your analysis.

24
Safety Stocks
Suppose that we assume orders occur at a fixed
review period and that demand is probabilistic
and we want a buffer stock to ensure that we
dont run out
Probability of stockout (1.0 - 0.85 0.15)
25
Safety Stock Formula
Reorder Point Average demand Safety stock
Reorder Point Demand during Lead Time Safety
Stock Demand during lead time daily demand L
dL Safety stock Zservicelevel sL Where
sL square root of Ls2, where s is the
standard deviation of demand for one day
26
Problem 4
  • A large manufacturer of VCRs sells 700,000 VCRs
    per year. Each VCR costs 100 and each time the
    firm places an order for VCRs the ordering charge
    is 500. The accounting department has
    determined that the cost of carrying a VCR for
    one year is 40 of the VCR cost. If we assume
    350 working days per year, a lead-time of 4 days,
    and a standard deviation of lead time of 20 per
    day, answer the following questions.
  • How many VCRs should the company order each time
    it places an order?
  • If the company seeks to achieve a 99 service
    level (i.e. a 1 chance of being out of stock
    during lead time), what will be the reorder
    point? How much lower will be the reorder point
    if the company only seeks a 90 service level?

27
Fixed-Time Period Model with Safety Stock Formula
q Average demand Safety stock Inventory
currently on hand
28
Multi-Period Models Fixed-Time Period Model
Determining the Value of sTL
  • The standard deviation of a sequence of random
    events equals the square root of the sum of the
    variances

29
Example of the Fixed-Time Period Model
Given the information below, how many units
should be ordered?
Average daily demand for a product is 20 units.
The review period is 30 days, and lead time is 10
days. Management has set a policy of satisfying
96 percent of demand from items in stock. At the
beginning of the review period there are 200
units in inventory. The daily demand standard
deviation is 4 units.
30
Example of the Fixed-Time Period Model Solution
(Part 1)
The value for z is found by using the Excel
NORMSINV function.
31
ABC Classification System
  • Items kept in inventory are not of equal
    importance in terms of
  • dollars invested
  • profit potential
  • sales or usage volume
  • stock-out penalties

So, identify inventory items based on percentage
of total dollar value, where A items are
roughly top 15 , B items as next 35 , and the
lower 65 are the C items
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